题目
题型:不详难度:来源:
在数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013832-76614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013832-19334.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013832-57067.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013833-41171.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013833-72956.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013833-68369.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013834-53492.png)
(2)猜想
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013832-76614.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013834-96839.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013834-90453.png)
解析
试题分析:(1)由条件得
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013834-39691.png)
由此可得
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013834-96839.png)
(2)猜测
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013834-90453.png)
用数学归纳法证明:
①当
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013835-27883.png)
②假设当
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013835-74391.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013835-88204.png)
那么当
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013835-18642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013836-46202.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013836-98175.png)
所以当
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013835-18642.png)
由①②可知,
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013834-90453.png)
对一切正整数都成立.
点评:数学归纳法证明的关键点在于由
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013836-11681.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013836-48027.png)
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013813-41562.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013814-89984.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013814-18720.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013814-21229.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013814-51516.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013814-89984.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013815-84817.png)
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013815-56624.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013815-74687.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013815-89808.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013815-97277.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013816-53631.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013815-89808.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013814-18720.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013814-89984.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013813-41562.png)
(3)是否存在这样的正数
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013814-18720.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013814-89984.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013814-18720.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013531-88608.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013532-25196.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013532-61320.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013532-84690.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013531-88608.png)
(2)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013533-91825.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013532-25196.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013533-27042.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013519-19953.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013519-84134.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013519-31239.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013519-79052.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013519-56117.png)
(Ⅰ)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013519-19953.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013520-99248.png)
(Ⅱ)若数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013520-44082.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013520-38370.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013520-98188.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013521-87224.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013519-84134.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013521-71208.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013502-64463.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013502-99105.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013502-73763.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013503-16279.png)
(1)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013502-64463.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013503-79773.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010013503-72179.png)
(2,2),(3,1),(1,4),(2,3),(3,2),(4,1),(1,5),
(2,4)…,则第57个数对是
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